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A CHEBYSHEV PSEUDO-SPECTRAL METHOD FOR SOLVING FRACTIONAL-ORDER INTEGRO-DIFFERENTIAL EQUATIONS

  • N. H. SWEILAM (a1) and M. M. KHADER (a2)
Abstract
Abstract

A Chebyshev pseudo-spectral method for solving numerically linear and nonlinear fractional-order integro-differential equations of Volterra type is considered. The fractional derivative is described in the Caputo sense. The suggested method reduces these types of equations to the solution of linear or nonlinear algebraic equations. Special attention is given to study the convergence of the proposed method. Finally, some numerical examples are provided to show that this method is computationally efficient, and a comparison is made with existing results.

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      A CHEBYSHEV PSEUDO-SPECTRAL METHOD FOR SOLVING FRACTIONAL-ORDER INTEGRO-DIFFERENTIAL EQUATIONS
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Corresponding author
For correspondence; e-mail: n˙sweilam@yahoo.com
References
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[1]Arikoglu A. and Ozkol I., “Solution of fractional integro-differential equations by using fractional differential transform method”, Chaos Solitons Fractals 34 (2009) 521529.
[2]Bagley R. L. and Torvik P. J., “A theoretical basis for the application of fractional calculus to viscoelasticity”, J. Rheol. 27 (1983) 201210.
[3]Baillie R. T., “Long memory processes and fractional integration in econometrics”, J. Econometrics 73 (1996) 559.
[4]Caputo M., “Linear models of dissipation whose Q is almost frequency independent. Part II”, Geophys. J. R. Astr. Soc. 13 (1967) 529539.
[5]Chow T. S., “Fractional dynamics of interfaces between soft-nanoparticles and rough substrates”, Phys. Lett. A 342 (2005) 148155.
[6]Constantinides A., Applied numerical methods with personal computers (McGraw-Hill, New York, 1987).
[7]Das S., Functional fractional calculus for system identification and controls (Springer, New York, 2008).
[8]Diethelm K., Ford N. J. and Luchko Yu., “Algorithms for the fractional calculus: a selection of numerical methods”, Comput. Methods Appl. Mech. Engrg. 194 (2005) 743773.
[9]Gorenflo R. and Vessella S., Abel integral equations (Springer, Berlin, 1991).
[10]Hashim I., Abdulaziz O. and Momani S., “Homotopy analysis method for fractional IVPs”, Commun. Nonlinear Sci. Numer. Simul. 14 (2009) 674684.
[11]Hussaini M. Y., Quarteroni A. and Zang T. A., Spectral methods in fluid dynamic (Prentice-Hall, Englewood Cliffs, NJ, 1988).
[12]Inc M., “The approximate and exact solutions of the space-and time-fractional Burger’s equations with initial conditions by variational iteration method”, J. Math. Anal. Appl. 345 (2008) 476484.
[13]Kadem A. and Baleanu D., “Fractional radiative transfer equation within Chebyshev spectral approach”, Comput. Math. Appl. 59 (2010) 18651873.
[14]Kadem A. and Baleanu D., “Analytical method based on Walsh function combined with orthogonal polynomial for fractional transport equation”, Commun. Nonlinear Sci. Numer. Simul. 15 (2010) 491501.
[15]Kilbas A. A., Srivastava H. M. and Trujillo J. J., Theory and applications of fractional differential equations (Elsevier, San Diego, 2006).
[16]Miller K. S. and Ross B., An introduction to the fractional calculus and fractional differential equations (John Wiley and Sons, New York, 1993).
[17]Momani S. and Noor M. A., “Numerical methods for fourth-order fractional integro-differential equations”, Appl. Math. Comput. 182 (2006) 754760.
[18]Oldham K. B. and Spanier J., Fractional calculus: theory and applications, differentiation and integration to arbitrary order (Academic Press, New York, 1974).
[19]Podlubny I., Fractional differential equations (Academic Press, San Diego, 1999).
[20]Rawashdeh E. A., “Numerical solution of fractional integro-differential equations by collocation method”, Appl. Math. Comput. 176 (2006) 16.
[21]Saadatmandi A. and Dehghan M., “Numerical solution of the one-dimensional wave equation with an integral condition”, Numer. Methods Partial Differential Equations 23 (2007) 282292.
[22]Saadatmandi A. and Dehghan M., “Numerical solution of a mathematical model for capillary formation in tumor angiogenesis via the tau method”, Comm. Numer. Methods Engrg. 24 (2008) 14671474.
[23]Snyder M.A., Chebyshev methods in numerical approximation (Prentice-Hall, Englewood Cliffs, NJ, 1966).
[24]Sweilam N. H., Khader M. M. and Al-Bar R. F., “Numerical studies for a multi-order fractional differential equation”, Phys. Lett. A 371 (2007) 2633.
[25]Yuanlu L., “Solving a nonlinear fractional differential equation using Chebyshev wavelets”, Commun. Nonlinear Sci. Numer. Simul. 15 (2010) 22842292.
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